An evolution problem for abstract differential equations is studied. The
typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad
\dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in
a Hilbert space $H$, and $F$ is a nonlinear operator, $\|F(t,u)\|\leq
c_0\|u\|^p,\,\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\leq
-\gamma(t)\|u\|^2$ $\forall u\in H$, where $\gamma(t)>0$, and the case when
$\lim_{t\to \infty}\gamma(t)=0$ is also considered. An estimate of the rate of
decay of solutions to problem (*) is given. The derivation of this estimate
uses a nonlinear differential inequality

Ramm, A. G.

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A. G. Ramm

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Asymptotic stability of solutions to abstract differential equations

Ramm, Alexander G.

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This is the author’s final, peer-reviewed manuscript as accepted for publication.  The publisher-formatted version may be available through the publisher’s web site or your institution’s library.  This item was retrieved from the K-State Research Exchange (K-REx), the institutional repository of Kansas State University.  K-REx is available at http://krex.ksu.edu    Asymptotic stability of solutions to abstract differential equations    A.G. Ramm  How to cite this manuscript  If you make reference to this version of the manuscript, use the following information:    Ramm, A.G. (2010). Asymptotic stability of solutions to abstract differential equations. Retrieved from http://krex.ksu.edu  Published Version Information   Citation:  Ramm, A.G. (2010). Asymptotic stability of solutions to abstract differential equations. Journal of Abstract Differential Equations and Applications,1(1), 27-34   Copyright: Copyright ©2010 Mathematical Research Publishers   Digital Object Identifier (DOI):     Publisher’s Link: http://math-res-pub.org/jadea/contents   Journal ofAbstractDifferentialEquations andApplicationsVolume 1, Number 1, pp. 27–34 (2010)ASYMPTOTIC STABILITY OF SOLUTIONS TO ABSTRACTDIFFERENTIAL EQUATIONSA. G. RAMM∗Kansas State University, Department of Mathematics,Manhattan, KS 66506-2602, USA(Communicated by Irena Lasiecka)Abstract. An evolution problem for abstract differential equations is studied. The typical problemis:u˙= A(t)u+F(t,u), t ≥ 0; u(0) = u0; u˙= dudt (∗)Here A(t) is a linear bounded operator in a Hilbert spaceH, and F is a nonlinear operator, ‖F(t,u)‖≤c0‖u‖p, p > 1, c0, p = const > 0. It is assumed that Re(A(t)u,u) ≤ −γ(t)‖u‖2 ∀u ∈ H, whereγ(t) > 0, and the case when limt→∞ γ(t) = 0 is also considered. An estimate of the rate of decayof solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differentialinequality.AMS Subject Classification: 34G20, 37L05, 44J05, 47J35.Keywords: Nonlinear inequality; Asymptotic stability; Abstract differential equations.Received: August 7, 2010 ‖ Accepted: September 27, 20101 IntroductionLetu˙(t) = A(t)u+F(t,u), t ≥ 0; u˙(t) := u˙= dudt, (1.1)u(0) = u0, (1.2)where u ∈ H, H is a Hilbert space, A(t) is a bounded linear operator in H, F(t,u) is a nonlinearoperator,‖F(t,u)‖ ≤ c0‖u‖p, p> 1, (1.3)c0 and p are positive constants, and u0 ∈ H.∗e-mail address: ramm@math.ksu.edu28 A. G. RammOne says that A(t) ∈ B(ρ,N) if every solution to the equationv˙(t) = A(t)v (1.4)satisfies the estimate‖v(t)‖ ≤ Neρ(t−s)‖v(s)‖, t ≥ s≥ 0, (1.5)where N > 0 and ρ are real numbers. This definition is discussed in [1] and goes back to P. Bohl(see the historical remarks in [1]). IfU(t,s) is an operator that solves the problemU˙(t,s) = A(t)U(t,s), t ≥ s; U(s,s) = I, (1.6)where I is the identity operator, then (1.5) is equivalent (by the Banach-Steinhaus theorem) to theestimate‖U(t,s)‖ ≤ Neρ(t−s), t ≥ s≥ 0. (1.7)Let us define, following [1], the notion of upper general exponent κ for the solutions to (1.4):κ = limt,s→∞ln‖U(t+ s,s)‖t, t,s≥ 0. (1.8)If κ < 0, then ‖v(t)‖= O(e−|κ|t) as t → ∞, s being fixed.The following result is obtained in [1], Theorem 3.1, Chapter 7.Proposition 1.1. If κ < 0 and assumption (1.3) holds, then the zero solution to equation (1.1) isasymptotically stable.Recall that the zero solution to equation (1.1) is called Lyapunov stable if for every ε > 0, onecan find a δ = δ (ε) > 0, such that if ‖u0‖ ≤ δ , then the solution to problem (1.1)−(1.2) satisfiesthe estimate supt≥0 ‖u(t)‖ ≤ ε . If, in addition, limt→∞ ‖u(t)‖= 0, then the zero solution to equation(1.1) is called asymptotically stable in the Lyapunov sense.As one can see from our proof of Theorem 1.2, the condition of smallness of the initial data‖u0‖ ≤ δ can be replaced by a different condition: if ‖u0‖ is arbitrary fixed, then one still derivesthe relation limt→∞ ‖u(t)‖= 0 from (2.8) (see below), provided that c0 is sufficiently small.In Proposition 1.1, the exponent κ < 0 is a constant. For example, if A(t) = A∗(t) is a selfadjointcompact operator, and λ j(t) are its eigenvalues, λ j(t) ≤ λm(t) < 0 if j > m, j = 1,2,3, ..., thenλ1(t)≤ κ < 0.Our goal is to derive an analog of Proposition 1.1 such that limt→∞λ1(t) = 0 is allowed, that is,we do not assume that the spectrum σ(A(t)) of A(t) lies in a half-plane Re z≤ κ , where κ < 0 is afixed constant independent of t.It is known (see, e.g., [1]) that if A is a bounded linear operator in H with the spectrum σ(A),which lies in the half-plane Re z ≤ −|κ|, |κ| > 0, then there is a positive-definite operatorW suchthat ReWA = −V , where V is an arbitrary given positive-definite operator in H. In other words, ifm= const > 0, σ(A)⊂{z : Rez≤−|κ|< 0} andV =V ∗ ≥m> 0, that is, (Vu,u)≥m(u,u) ∀u∈H,then the operator equation A∗W +WA=−2V is solvable forW . In fact, there is an explicit formulaforW : W = 2∫ ∞0 eA∗tVeAtdt (see [1]). By ReA one understands the operator defined by the formulaReA := AR := (A+A∗)/2, and A = AR + iAI , where AR and AI are selfadjoint operators that arecalled the real and imaginary parts of A. If AR ≤−a, a= const > 0, then σ(A) lies in the half-planeRe z≤−a. The notation A≤−a means (Au,u)≤−a(u,u) ∀u ∈ H.The converse is not true: it is not true that if the spectrum of a linear bounded operator Alies in the half-plane Re z ≤ −a, then the inequality AR ≤ −a holds. A simple counterexample isAsymptotic Stability of Solutions to Abstract Differential Equations 29given by the following 2×2 matrix A in R2, A=(0 b−a −1). The eigenvalues of this matrix are−0.5± i√ab−0.25, and if ab≥ 0.25, then the spectrum σ(A) of A lies in the half-plane Rez≤−0.5.On the other hand, if, for example, a= 1 and b= 5, u1 = u2 = 0.5, then (ARu,u)> 0.Inequality Re(Au,u)≤ 0 means that the operator A(t) is dissipative. Such operators often arisein applications (see, e.g., [9]). The dissipativity property, defined by the above inequality, usuallymeans that the energy in the system is dissipating, that is, the system is passive. In [7] a wide classof passive nonlinear networks is studied, see also [8], Chapter 3.Our basic results on the stability of the solutions to problem (1.1)-(1.2) with dissipative operatorA(t) are formulated in Theorems 1.2 and 1.4. Theorem 1.2 contains an auxiliary result used in theproofs of Theorems 1.2 and 1.4. This result is of interest by itself and useful in applications.Theorem 1.2. Assume that Re(Au,u)≤−|κ|‖u‖2 for every u ∈H and inequality (1.3) holds. Thenthe solution to problem (1.1)−(1.2) satisfies an estimate ‖u(t)‖ = O(e−(|κ|−ε)t) as t → ∞. Here0< ε < |κ| can be chosen arbitrarily small if ‖u0‖ is sufficiently small.This theorem implies asymptotic stability in the sense of Lyapunov of the zero solution to equa-tion (1.1). Our proof of Theorem 1.2 is new and very short.We first prove Theorem 1.2 and Theorem 1.4 in Section 2, because the ideas of our proofs ofthese theorems are quite similar. Theorem 1.4 contains a new result, and it is not assumed in theformulation of this theorem that the spectrum of A(t) lies in a half-plane Rez ≤ −|κ| with |κ| > 0being a constant independent of t.Then we prove Theorem 1.3. The result of this theorem is used in the proofs of Theorems1.2 and 1.4, and is of general interest. It gives a bound on solutions to a nonlinear differentialinequality. Results of this type, but considerably less general, were used extensively in [6], where theDynamical Systems Method (DSM) for solving operator equations, especially nonlinear equations,was developed.The ideas of our proofs are quite different from these in [1].Theorem 1.3. Let g(t)≥ 0 be defined on an interval [0,T ), T > 0, and have a bounded derivativefrom the right at every point of this interval, g˙(t) := lims→+0g(t+s)−g(t)s . Assume that g(t) satisfiesthe following inequalityg˙(t)≤−γ(t)g(t)+α(t,g(t))+β (t), t ∈ [0,T ); g(0) = g0, (1.9)where β (t)≥ 0 and γ(t)≥ 0 are continuous functions, defined on [0,∞), and α(t,v)≥ 0 is definedon [0,∞)× [0,∞), α(t,v) is non-decreasing as a function of v, locally Lipschitz with respect to v,and continuous with respect to t on R+ := [0,∞).If there exists a function µ > 0, continuously differentiable on R+, such thatα(t,1µ(t))+β (t)≤ 1µ(t)(γ(t)− µ˙(t)µ(t)), ∀t ≥ 0, (1.10)andg(0)<1µ(0), (1.11)then g(t) exists for all t ≥ 0, that is, g(t) can be extended from [0,T ) to [0,∞), and g(t) satisfies thefollowing inequality:0≤ g(t)< 1µ(t), ∀t ≥ 0. (1.12)If g(0)≤ 1µ(0) , then 0≤ g(t)≤ 1µ(t) , ∀t ≥ 0.30 A. G. RammInequality (1.12) was formulated in [5] under some different assumptions, but not proved there.We sketch its proof at the end of this paper.In [4] inequality (1.9) is studied in the case that includes α(t,g) = c0gp, where p> 1 and c0 > 0are constants, as a particular case: the coefficient c0 in [4] was a function of time.Our second stability result is the following theorem.Theorem 1.4. Assume that inequality (1.3) holds,Re(A(t)u,u)≤−γ(t)‖u‖2, ∀t ≥ 0, (1.13)andγ(t) =c1(1+ t)q, q≤ 1; c1,q= const > 0. (1.14)Suppose that ε ∈ (0,c1) is an arbitrary fixed number, λ =( c0ε)1/(p−1), (p− 1)(c1− ε) ≥ q, and‖u(0)‖ ≤ 1λ .Then the unique solution to (1.1)−(1.2) exists on all of R+ and0≤ ‖u(t)‖ ≤ 1λ (1+ t)c1−ε. (1.15)Remark 1. One may change the formulation of Theorem 1.4 as follows: if for some positiveconstants λ and ν > 0 inequalities (2.12) and (2.7) (see below) hold, then inequality ‖u(t)‖ ≤1λ (1+t)ν holds for all t ≥ 0 for the solution to problem (1.1)−(1.2), as follows from the proof ofTheorem 1.4, given in Section 3.The rate of decay of the solution u(t) as t → ∞, obtained in Theorem 1.4, is not necessarily thebest possible. The result in Theorem 1.4 is novel and interesting because no assumption of the typeγ(t) ≥ γ0 > 0, where γ0 is a constant, is made. This allows one to study, for instance, evolutionproblems with elliptic operators A(t) the ellipticity constant λ (t) of which may tend to zero ast → ∞. Here λ (t) is the smallest eigenvalue of the matrix ai j(t) of the elliptic operator A(t). Anexample is given in Remark 2, at the end of the paper.We have assumed above that A(t) is a bounded linear operator, since this assumption is basic inthe book [1], and in the Introduction to our paper a comparison was made with the results in [1].However, boundedness of A(t) was not used in our arguments. If A(t) is a bounded linear operatorsatisfying the assumptions of Theorems 1.2 or 1.4, then one can guarantee the global existence ofthe solution to evolution problem (1.1)−(1.2). If A(t) is an unbounded linear operator for which theglobal existence of u(t) holds, then our arguments, which lead to estimate (1.15), remain valid. Inthe example given in Remark 2, the operator A(t) = γ(t)(∆− I), where ∆ is a selfadjoint realizationof the Laplacian in H = L2(R3), and I is the identity operator in H. For this A(t) one knows that thesolution u(t) to problem (1.1)−(1.2) exists globally, so Theorem 1.4 is applicable.In Section 2 proofs are given.2 ProofsProof. (Proof of Theorem 1.2).Multiply (1.1) by u, denote g= g(t) := ‖u(t)‖, take the real part, and use the assumption (1.13)with γ(t) = |κ|= const > 0, to getgg˙≤−|κ|g2+ c0gp+1, p> 1. (2.1)Asymptotic Stability of Solutions to Abstract Differential Equations 31If g(t)> 0 then the derivative g˙ does exist, as one can easily check. If g(t) = 0 on an open subset ofR+, then the derivative g˙ does exist on this subset and g˙(t) = 0 on this subset. If g(t) = 0 but in anyneighborhood (t−δ , t+δ ) there are points at which g does not vanish, then by g˙ we understand thederivative from the right, that is,g˙(t) := lims→+0g(t+ s)−g(t)s= lims→+0g(t+ s)s.This limit does exist and is equal to ‖u˙(t)‖. Indeed, the function u(t) is continuously differentiable,solims→+0‖u(t+ s)‖s= lims→+0‖u˙(t)+o(1)‖= ‖u˙(t)‖.The assumption about the existence of the bounded derivative g˙(t) from the right in Theorem 1.3was made because the function ‖u(t)‖ does not have, in general, a derivative in the usual sense atthe points τ at which ‖u(τ)‖= 0, no matter how smooth the function u(t) is at the point τ . However,as we have proved above, the derivative g˙(t) from the right does exist always, if u(t) is continuouslydifferentiable at the point t.Since g ≥ 0, the inequality (2.1) yields inequality (1.9) with γ(t) = |κ| = const > 0, β (t) = 0,and α(t,g) = c0gp. Inequality (1.10) takes the formc0µ p(t)≤ 1µ(t)(|κ|− µ˙(t)µ(t)), ∀t ≥ 0. (2.2)Letµ(t) = λebt , λ ,b= const > 0, (2.3)and choose the constants λ and b later. Then inequality (2.2) takes the formc0λ p−1e(p−1)bt+b≤ |κ|, ∀t ≥ 0. (2.4)This inequality holds ifc0λ p−1+b≤ |κ|. (2.5)Let ε > 0 be an arbitrary small fixed number. Choose b= |κ|− ε > 0. Then (2.5) holds ifλ ≥ (c0ε) 1p−1 . (2.6)Condition (1.11) holds if‖u0‖= g(0)≤ 1λ . (2.7)From (2.6), (2.7) and (1.12) one gets0≤ g(t) = ‖u(t)‖ ≤ e−(|κ|−ε)tλ, ∀t ≥ 0. (2.8)Theorem 1.2 is proved.Proof. (Proof of Theorem 1.4.)We start with inequality (2.2), letµ(t) = λ (1+ t)ν , λ ,ν = const > 0, (2.9)32 A. G. Rammand choose the constants λ and ν later. Inequality (2.2) holds ifc0λ p−1(1+ t)(p−1)ν+ν1+ t≤ c1(1+ t)q, ∀t ≥ 0. (2.10)Ifq≤ 1, (p−1)ν ≥ q, (2.11)then inequality (2.10) holds ifc0λ p−1+ν ≤ c1. (2.12)Let ε > 0 be an arbitrary small number. Chooseν = c1− ε. (2.13)Then (2.12) holds if (2.6) holds. Inequality (1.11) holds if (2.7) holds. Combining (2.6), (2.7) and(1.12), one obtains0≤ ‖u(t)‖= g(t)≤ 1λ (1+ t)c1−ε, ∀t ≥ 0. (2.14)Choose λ =( c0ε) 1p−1 . Then inequality (2.12) holds because of (2.13). Inequality (1.11) holds be-cause we have assumed in Theorem 1.4 that ‖u(0)‖ ≤ 1λ . Thus, the desired inequality (1.15) holdsby Theorem 1.3.Theorem 1.4 is proved.Proof. (Proof of Theorem 1.3.)Definev(t) := g(t)a(t), a(t) := e∫ tt0γ(s)ds, η(t) :=a(t)µ(t). (2.15)Then inequality (1.9) takes the formv˙(t)≤ a(t)[α(t,v(t)a(t))+β (t)], v(0) = g(0) := g0, (2.16)andη˙(t) =a(t)µ(t)[γ(t)− µ˙(t)µ(t)]. (2.17)From inequalities (1.11) and (1.10) one getsv(0)<1µ(0)= η(0), v˙(0)≤ η˙(0). (2.18)Thus, v(t)< η(t) on some interval [0,T ]. Inequalities (2.16), (2.17), and (1.10) implyv˙(t)≤ η˙(t), t ∈ [0,T ]. (2.19)It follows from inequalities (2.18) and (2.19) thatv(t)< η(t), ∀t ≥ 0. (2.20)From inequalities (2.20) and (2.15) one obtainsa(t)g(t) = v(t)< η(t) =a(t)µ(t), ∀t ≥ 0. (2.21)Asymptotic Stability of Solutions to Abstract Differential Equations 33Since a(t) > 0, inequality (2.21) is equivalent to inequality (1.12). This essentially completes themajor part of the proof of inequality (1.12). The last conclusion of Theorem 1.3 can be obtained bya standard limiting procedure.Let us explain in detail why inequality (2.21) holds for all t ≥ 0. The right-hand side of inequal-ity (2.21) is defined for all t ≥ 0. The function g(t), a solution to inequality (1.9), exists on everyinterval on which v(t) exists, and v(t), the solution to inequality (2.16), exists on every interval onwhich the solution w(t) to the problemw˙(t) = a(t)[α(t,w(t)a(t)+β (t)], w(0) = v(0) (2.22)exists. It follows from inequality (2.16) and equation (2.22) that v(t)≤ w(t) on every interval [0,T )on which w exists. We have already proved that the solution to problem (2.22) (which also is asolution to problem (2.16)) satisfies the estimate0≤ w(t)≤ a(t)µ(t)(2.23)on every interval on which w exists. We claim that estimate (2.23) implies that w exists for all t ≥ 0,in other words, that T = ∞. Indeed, according to the known result (see, e.g., [2], Theorem 3.1 inChapter 2), if the maximal interval [0,T ) of the existence of the solution to problem (2.22) is finite,that is T < ∞, then limt→T−0w(t) = ∞. This, however, cannot happen because of the inequality(2.23), since the function a(t)µ(t) is bounded for every t ≥ 0.Theorem 1.3 is proved.Remark 2. Let H = L2(R3), A(t) = γ(t)A, where A = A∗ is a selfadjoint operator in H whichis the closure of a symmetric operator ∆− I with the domain of definition C∞0 (R3). Here ∆ isthe Laplacian. Let γ(t) be defined in (1.14) with c1 = 1, q = 0.5. let ε = 0.01, p = 3, c0 =1, λ = 10, ν = 0.99. Assume that ‖u0‖ ≤ (0.99)−1. Theorem 1.4 yields the following estimate‖u(t)‖ ≤ 0.1(1+ t)−0.99 for the solution u(t) to problem (1.1)−(1.2) with the defined above A(t)and a nonlinearity satisfying condition (1.3).References[1] Yu. L. Daleckii, M. G. Krein, Stability of solutions of differential equations in Banach spaces, Amer.Math. Soc., Providence, RI, 1974.[2] P. Hartman,Ordinary differential equations, J.Wiley, New York, 1964.[3] N. S. Hoang and A. G. Ramm, A nonlinear inequality, Jour. Math. Ineq., 2, N4, (2008), 459-464.[4] N. S. Hoang and A. G. Ramm, A nonlinear inequality and applications, Nonlinear Analysis: Theory,Methods & Applications, 71, (2009), 2744- 2752.[5] N. S. Hoang and A. G. Ramm, DSM of Newton-type for solving operator equations F(u) = f withminimal smoothness assumptions on F , International Journ. Comp.Sci. and Math. (IJCSM), 3, N1/2,(2010), 3-55.[6] A. G. Ramm, Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 2007.[7] A. G. Ramm, Stationary regimes in passive nonlinear networks, in the book “Nonlinear Electromag-netics”, Editor P. Uslenghi, Acad. Press, New York, 1980, pp. 263-302.34 A. G. Ramm[8] A. G. Ramm, Theory and applications of some new classes of integral equations, Springer-Verlag, NewYork, 1980.[9] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer-Verlag, NewYork, 1997.

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Asymptotic stability of solutions to abstract differential equations

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